Essential German Grammar (good book) by Guy Stern and Everett F. Bleiler has this to say about using 'sein' as an auxiliary verb:

Only three categories of German verbs use sein as an auxiliary. These are:

1. Verbs involving a change of position that cannot take a direct object.
   [Table showing 'kommen', 'gehen', 'fahren', 'fallen', 'steigen', and 'folgen'.]


2. Verbs involving a change of condition that cannot take an object.
   [Table showing 'wachsen', 'sterben', and 'verschwinden'.]


3. Miscellaneous verbs.

Yeah, and there are only three types of words: verbs, nouns, and all the other types.

If one of your categories is just a catch-all, your classification system is incomplete. There's only one category of X that Y: 'all X that Y'.

(I can't think of any sensible labels[==categories] for this post.)